On the Levelt’s Theorem

Abstract

Let (E) be a homogeneous linear differential equation Fuchsian of order n over P1 (C). The idea of Riemann (1857) was to obtain the properties of solutions of (E) by studying the local system. Thus, he obtained some properties of Gauss hypergeometric functions by studying the associated rank 2 local system over P1 (C) \ {3 points}. For example, he obtained the Kummer transformations of the hypergeometric functions without any calculation. The success of the Riemann’s methods is due to the fact that the irreducible rank 2 local system over P1 (C) \ {3 points} is linearly ”rigid” in the sense of Katz. This result constitute one of the best studied example of linear rigid system, it was proved by the Levelt’s theorem. In this work we propose a partial generalization of this theorem.